ALBANY — The corruption scandal enveloping the Nassau County executive has given Democrats a new and potent line of attack, as they try to gain control of the narrowly divided New York Senate.

The executive, Edward P. Mangano, a Republican, was arrested by federal agents on Thursday along with his wife, Linda Mangano, and the Oyster Bay town supervisor, John Venditto, also a Republican. All were charged with a scheme that included government contracts, bribery, kickbacks and a no-show job involving food tasting.

The charges sent ripples through the county on Long Island, where Mr. Mangano wielded much influence through a warm relationship with members of the Republican establishment as well as with Gov. Andrew M. Cuomo, a Democrat whom he endorsed across party lines in 2014. (Mr. Cuomo has had no comment on the arrest.)

Mr. Mangano’s arrest also has many Democratic leaders dreaming of a countywide sweep, noting that their party enjoys a slight advantage in voter registration and that they hope to use the corruption scandal in tandem with efforts to tie Republican candidates to Donald J. Trump, who they believe has alienated moderate suburban voters.

“Nassau County was a focal point for us this fall just because the demographics and the presidential race were making it competitive,” said Senator Michael N. Gianaris of Queens, the Senate Democrats’ chief political strategist. “And with all our candidates stressing ethics reform and corruption as our main message, recent events make it more favorable than ever.”

There are perhaps 10 competitive races statewide that could tip the balance of power in the Senate, where the math of majority can be hard to follow: The 63-seat chamber actually includes 32 Democrats, but one Democrat — Senator Simcha Felder of Brooklyn — caucuses with the Republicans, giving that party power.

Democrats hope to pick up at least one seat to seize the majority, but even so, they also will probably have to work with the Independent Democratic Conference, a breakaway group of Democratic senators who have often worked closely with Republicans.

But the Democrats are aiming larger, focusing on seemingly safe Republican seats like the Eighth District on the South Shore of Long Island. But because that district is held by Senator Michael Venditto, the son of John Venditto, Democrats feel that the electoral map has expanded.

On Thursday, Democrats began new internal polling in the Eighth District on behalf of their candidate, John E. Brooks, to determine a path to victory. And on Friday, Mr. Brooks unveiled a four-point ethics plan, saying that “corrupt officials have lined their pockets at the expense of Long Island taxpayers, and people like Senator Mike Venditto have stood by and let it happen.”

As Mr. Gianaris suggested, the issue of corruption had already been in play in several Nassau races, including the Ninth District, which was touched by scandal last year as well: It is the former seat of Dean G. Skelos, a Republican who was the Senate majority leader before being convicted of federal corruption charges in December. The incumbent, Todd Kaminsky, a Democrat who won a special election in April, is a former federal prosecutor, which has been highlighted in his campaign ads.

Mr. Kaminsky wasted little time calling for Mr. Mangano and John Venditto’s resignations, along with a “wholesale ethics reform.”

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State Senator Kemp Hannon, right, a Republican, is running for re-election in the Sixth District. He is facing Ryan Cronin, whom he beat in a 2012 race by fewer than 5,000 votes.CreditNathaniel Brooks for The New York Times

“Pay-to-play politics is disgraceful,” Mr. Kaminsky said in a statement released on Thursday. “And taxpayers deserve better.”

That call was echoed by every other Democratic Senate candidate in the county — each used the phrase “resign immediately” — and placed Republicans in an awkward position of doing the same. This included Mr. Kaminsky’s opponent, Chris McGrath, a lawyer and Hofstra University law professor, who said in an interview that “the public deserves someone who will give 100 percent of their time to the job.”

Republican officials noted that all five of their candidates also called for resignations and denounced the alleged behavior outlined by Robert L. Capers, the United States attorney for the Eastern District of New York.

“We have zero tolerance for corruption of any kind, regardless of whether it is committed by a Democrat or a Republican,” said Scott Reif, a spokesman for the Senate Republicans. “Over the next few weeks, we are going to remind people of the comprehensive ethics reforms we have already achieved, and that we are committed to doing more to ensure citizens have faith and trust in their government at every level.”

Among those distancing themselves from Mr. Mangano and Mr. Venditto was Carl L. Marcellino, a Republican whose district includes parts of the more conservative Suffolk County and who is in a battle with James Gaughran, a Democrat. In a Facebook post on Thursday, Senator Marcellino also expressed doubts about Mr. Venditto and Mr. Mangano’s ability to “effectively lead the county or town and represent the people in the manner they deserve with this cloud over their heads.”

Mr. Gaughran, the chairman of the Suffolk County Water Authority, had already tried to make ethics a part of the race, with campaign mailers arriving this week that characterize Mr. Marcellino, who has deep ties to Oyster Bay, as part of a “corrupt political machine.”

Democrats are also optimistic about the Sixth District, where the Republican incumbent, Kemp Hannon, is facing Ryan Cronin in a rematch of a 2012 race. Senator Hannon won that race by fewer than 5,000 votes.

In the Seventh District, the incumbent senator, Jack M. Martins, is leaving to run for Congress. The resulting race pits Adam Haber, a Nassau businessman whom Senator Martins handily defeated in 2014, against Elaine Phillips, the mayor of the village of Flower Hill, on the North Shore.

Craig Burnett, an assistant professor of political science at Hofstra who has studied the Senate races, said that Mr. Marcellino and Mr. Venditto’s seats should be safely Republican, but that the arrests could complicate that, even if the incumbents are state officials and not county. “Voters don’t necessarily make these distinctions,” Professor Burnett said, adding, “They know that there’s people on the ballot and that’s the only way they have to express themselves.”

He added that Mr. Kaminsky’s victory in April over Mr. McGrath in the Ninth District showed that solid Republican seats could flip. “There is evidence there that this can be done,” he said.

Professor Burnett and others said the younger Mr. Venditto, a first-term legislator, could lose votes from those who confuse him with his recently arrested father. Mr. Venditto offered an emotional statement on Thursday about the arrest, calling it “an extremely difficult situation for our entire family.”

“It is breaking our hearts,” he said in the statement, noting that John Venditto should be considered innocent until proven guilty. “I love my father.”

A version of this article appears in print on , on Page A20 of the New York edition with the headline: Nassau Official’s Arrest Raises Democratic Hopes of Gains in State Senate. Order Reprints | Today’s Paper | Subscribe